Finite Group Databasebeta

← C109×#10243 #42128237

Subgroups — C109×#10243 #42128237

Order 27904.

578 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3226C211yesmaximal, characteristic
2C109×#3037C211yesmaximal, characteristic
3C109×#2546C211yesmaximal, characteristic, complemented
4C109×#1546C211yesmaximal, characteristic, complemented
5C109×#1502C211yesmaximal, characteristic
6C109×#1479C211yesmaximal, characteristic, complemented
7C109×#1303C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109—21no—
11C2×C4⋊Q8×C109C2211yescharacteristic
12C2×C4×Q8×C109C2211yescharacteristic
13#500×C109—21no—
14#486×C109—21no—
15#480×C109—22no—
16#474×C109—22no—
17#464×C109—22no—
18#462×C109—21no—
19#460×C109—21no—
20#431×C109C2211yescharacteristic
21#431×C109C2211yescharacteristic
22#423×C109C2211yescharacteristic
23#417×C109—21no—
24#398×C109—22no—
25#391×C109—21no—
26#389×C109—22no—
27C4×C4⋊C4×C109C2211yescharacteristic
28#332×C109—22no—
29#327×C109—22no—
30Q8⋊C8×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33C22×Q8×C109D411yescharacteristic
34#137×C109—42no—
35#137×C109—42no—
36C2×Q16×C109—22no—
37C2×SD16×C109—22no—
38C2×M16×C109—21no—
39C4⋊Q8×C109—21no—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—42no—
44C4⋊Q8×C109—42no—
45C4⋊Q8×C109D411yescharacteristic
46#127×C109D411yescharacteristic
47#124×C109—21no—
48#121×C109—21no—
49#121×C109—21no—
50C4×Q8×C109—21no—
51C4×Q8×C109—21no—
52C4×Q8×C109—21no—
53C4×Q8×C109—21no—
54C4×D4×C109—21no—
55C4×D4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109—24no—
58C2×C4⋊C4×C109—21no—
59C2×C4⋊C4×C109—21no—
60C2×C4⋊C4×C109D411yescharacteristic
61C2×C4⋊C4×C109D412yes—
62C2×C42×C109—21no—
63C2×C42×C109—21no—
64C2×C42×C109C2311yescharacteristic
65#107×C109—24no—
66C4⋊C8×C109—21no—
67C4⋊C8×C109—21no—
68C4⋊C8×C109D412yes—
69C4⋊C8×C109D412yes—
70#103×C109—22no—
71#102×C109—22no—
72#102×C109—24no—
73C22⋊C8×C109—21no—
74C22⋊C8×C109—21no—
75C4×C8×C109—21no—
76C4×C8×C109—21no—
77#95×C109—24no—
78#95×C109D412yes—
79#41×C109—41no—
80#41×C109—41no—
81#41×C109—41no—
82#41×C109—41no—
83C2×Q8×C109—21no—
84C2×Q8×C109—21no—
85C2×Q8×C109—21no—
86C2×Q8×C109—21no—
87C2×Q8×C109—21no—
88C2×Q8×C109—21no—
89C2×Q8×C109—41no—
90C2×Q8×C109—41no—
91C2×Q8×C109—41no—
92C2×Q8×C109—41no—
93C2×Q8×C109—41no—
94C2×Q8×C109—41no—
95C2×Q8×C109D812yes—
96C2×D4×C109—21no—
97C2×D4×C109—21no—
98C2×D4×C109—21no—
99C2×D4×C109—41no—
100C22×C4×C109—22no—
101C22×C4×C109—21no—
102C22×C4×C109—21no—
103C22×C4×C109—22no—
104C22×C4×C109—41no—
105C22×C4×C109C2×D411yescharacteristic
106C22×C4×C109C2×D411yescharacteristic
107C22×C4×C109C2×D411yescharacteristic
108Q16×C109—44no—
109SD16×C109—44no—
110M16×C109—41no—
111M16×C109—41no—
112C2×C8×C109—22no—
113C2×C8×C109—22no—
114C2×C8×C109—41no—
115C4⋊C4×C109—24no—
116C4⋊C4×C109—21no—
117C4⋊C4×C109—22no—
118C4⋊C4×C109—21no—
119C4⋊C4×C109—21no—
120C4⋊C4×C109—22no—
121C4⋊C4×C109—21no—
122C4⋊C4×C109—48no—
123C4⋊C4×C109—41no—
124C4⋊C4×C109—41no—
125C4⋊C4×C109—41no—
126C4⋊C4×C109—41no—
127C22⋊C4×C109—41no—
128C22⋊C4×C109—41no—
129C42×C109—41no—
130C42×C109—41no—
131C42×C109—41no—
132C42×C109—41no—
133C42×C109C2×D411yescharacteristic
134C42×C109C2×D411yescharacteristic
135C42×C109C2×D411yescharacteristic
136C42×C109#4111yescharacteristic
137C23×C109—41no—
138C23×C109#12011yescharacteristic
139Q8×C109—22no—
140Q8×C109—22no—
141Q8×C109—41no—
142Q8×C109—41no—
143Q8×C109—41no—
144Q8×C109—41no—
145Q8×C109—81no—
146Q8×C109—81no—
147Q8×C109—81no—
148Q8×C109—81no—
149D4×C109—41no—
150D4×C109—41no—
151D4×C109—42no—
152D4×C109—81no—
153D4×C109—81no—
154D4×C109—81no—
155C2×C4×C109—22no—
156C2×C4×C109—24no—
157C2×C4×C109—22no—
158C2×C4×C109—41no—
159C2×C4×C109—42no—
160C2×C4×C109—41no—
161C2×C4×C109—42no—
162C2×C4×C109—41no—
163C2×C4×C109—42no—
164C2×C4×C109—41no—
165C2×C4×C109—41no—
166C2×C4×C109—42no—
167C2×C4×C109—42no—
168C2×C4×C109—42no—
169C2×C4×C109—81no—
170C2×C4×C109—81no—
171C2×C4×C109#12011yescharacteristic
172C2×C4×C109#12011yescharacteristic
173C2×C4×C109#12111yescharacteristic
174C2×C4×C109#12111yescharacteristic
175C2×C4×C109#12111yescharacteristic
176C2×C4×C109#12711yescharacteristic
177C2×C4×C109C2×D811yescharacteristic
178C2×C4×C109#13511yescharacteristic
179C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
180C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
181C8×C109—42no—
182C8×C109—42no—
183C8×C109—81no—
184C22×C109—21no—
185C22×C109—21no—
186C22×C109—81no—
187C22×C109—81no—
188C22×C109—81no—
189C22×C109#39211yescharacteristic
190C22×C109#44711yescharacteristic
191C22×C109#44911yescharacteristic
192C4×C109—21no—
193C4×C109—21no—
194C4×C109—21no—
195C4×C109—44no—
196C4×C109—42no—
197C4×C109—82no—
198C4×C109—81no—
199C4×C109—81no—
200C4×C109#44711yescharacteristic
201C4×C109#44911yescharacteristic
202C4×C109#45612yes—
203C4×C109#46611yescharacteristic
204C4×C109#46911yescharacteristic
205#10243C10911yesmaximal, characteristic, direct product factor, Sylow
206C2×C109—21no—
207C2×C109—21no—
208C2×C109—161no—
209C2×C109#199911yescharacteristic
210C2×C109#200511yescharacteristic
211C2×C109#200911yescharacteristic
212#3226C2×C10911yescharacteristic
213#3037C2×C10911yescharacteristic
214#2546C2×C10911yescharacteristic, complemented
215#1546C2×C10911yescharacteristic, complemented
216#1502C2×C10911yescharacteristic
217#1479C2×C10911yescharacteristic, complemented
218#1303C2×C10911yescharacteristic, complemented
219C109#1024311yescharacteristic, direct product factor, Sylow
220C2×#137—21no—
221#532C22×C10911yescharacteristic
222C2×C4⋊Q8—21no—
223C2×C4⋊Q8C22×C10911yescharacteristic
224C2×C4×Q8C22×C10911yescharacteristic
225#500—21no—
226#486—21no—
227#480—22no—
228#474—22no—
229#464—22no—
230#462—21no—
231#460—21no—
232#431C22×C10911yescharacteristic
233#431C22×C10911yescharacteristic
234#423C22×C10911yescharacteristic
235#417—21no—
236#398—22no—
237#391—21no—
238#389—22no—
239C4×C4⋊C4C22×C10911yescharacteristic
240#332—22no—
241#327—22no—
242Q8⋊C8—22no—
243C2×#41—21no—
244C22×Q8—21no—
245C22×Q8D4×C10911yescharacteristic
246#137—42no—
247#137—42no—
248C2×Q16—22no—
249C2×SD16—22no—
250C2×M16—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—42no—
256C4⋊Q8—42no—
257C4⋊Q8D4×C10911yescharacteristic
258#127D4×C10911yescharacteristic
259#124—21no—
260#121—21no—
261#121—21no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×D4—21no—
267C4×D4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D4×C10911yescharacteristic
273C2×C4⋊C4D4×C10912yes—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C23×C10911yescharacteristic
277#107—24no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8D4×C10912yes—
281C4⋊C8D4×C10912yes—
282#103—22no—
283#102—22no—
284#102—24no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289#95—24no—
290#95D4×C10912yes—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8D8×C10912yes—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—41no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—41no—
317C22×C4C2×D4×C10911yescharacteristic
318C22×C4C2×D4×C10911yescharacteristic
319C22×C4C2×D4×C10911yescharacteristic
320Q16—44no—
321SD16—44no—
322M16—41no—
323M16—41no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C4⋊C4—24no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—21no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42C2×D4×C10911yescharacteristic
346C42C2×D4×C10911yescharacteristic
347C42C2×D4×C10911yescharacteristic
348C42#41×C10911yescharacteristic
349C23—41no—
350C23#120×C10911yescharacteristic
351Q8—22no—
352Q8—22no—
353Q8—41no—
354Q8—41no—
355Q8—41no—
356Q8—41no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—81no—
365D4—81no—
366D4—81no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#120×C10911yescharacteristic
384C2×C4#120×C10911yescharacteristic
385C2×C4#121×C10911yescharacteristic
386C2×C4#121×C10911yescharacteristic
387C2×C4#121×C10911yescharacteristic
388C2×C4#127×C10911yescharacteristic
389C2×C4C2×D8×C10911yescharacteristic
390C2×C4#135×C10911yescharacteristic
391C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
392C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
393C8—42no—
394C8—42no—
395C8—81no—
396C22—21no—
397C22—21no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#392×C10911yescharacteristic
402C22#447×C10911yescharacteristic
403C22#449×C10911yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—44no—
408C4—42no—
409C4—82no—
410C4—81no—
411C4—81no—
412C4#447×C10911yescharacteristic
413C4#449×C10911yescharacteristic
414C4#456×C10912yes—
415C4#466×C10911yescharacteristic
416C4#469×C10911yescharacteristic
417C2—21no—
418C2—21no—
419C2—161no—
420C2C109×#199911yescharacteristic
421C2C109×#200511yescharacteristic
422C2C109×#200911yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radical#532×C109C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C109×#2546C2semidirect
4. C109×#1546C2semidirect
6. C109×#1479C2semidirect
7. C109×#1303C2semidirect
205. #10243C109direct (N × H)
214. #2546C2×C109semidirect
215. #1546C2×C109semidirect
217. #1479C2×C109semidirect
218. #1303C2×C109semidirect
219. C109#10243direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C109×#32262
C109×#30372
C109×#25462
C109×#15462
C109×#15022
C109×#14792
C109×#13032
#10243109